Fourier's law and heat equation

1. Fourier’s Law
and the
Heat Equation
Chapter Two

2. Recap Chapter 1:
• Conduction heat transfer is governed by Fourier’s law.
• Determination of heat flux depends variation of temperature within the medium.
• One-dimensional, steady state conduction in a plane wall.
Chapter 2:
Objectives
• Application of Fourier’s law in different geometrical objects.
• Develop heat equation that gives temperature distribution.

3. Fourier’s Law
• A rate equation that allows determination of the conduction heat flux
from knowledge of the temperature distribution in a medium
Fourier’s Law
• Its most general (vector) form for multidimensional conduction is:
q k T

   
Implications:
– Heat transfer is in the direction of decreasing temperature
(basis for minus sign).
– Direction of heat transfer is perpendicular to lines of constant
temperature (isotherms).
– Heat flux vector may be resolved into orthogonal components.
– Fourier’s Law serves to define the thermal conductivity of the
medium /k q T
 
   
 

4. 4
GENERAL HEAT CONDUCTION
EQUATION
Some comments and points to remember:
• Temperature is a scalar, it only has magnitude
• Heat transfer is a vector quantity, is has magnitude and
direction
• Generally accepted that heat transfer is positive in the
positive direction of the coordinate system used.
• ΔT is the driving force for heat transfer
• In some problems, it is the determination of this ΔT that is
required. This ΔT may vary with spatial location
• Need to first specify which coordinate system is to be used
to specify a location in space or in a medium.
Can use:
Rectangular / Cartesian (x, y, z)
Cylindrical (r,φ,z)
Spherical (r,φ,θ) (φ - azimuthal; θ – zenith)

5. 5
x= r cos φ, y= r sin φ,
z = z
x= r cosφ sinθ ; y = r sinφ sinθ;
z = r cos θ
x, y, z
Rectangular coordinates
(box, room)
Cylindrical coordinates
(pipe)
Spherical coordinates
(container, tank)

6. Heat Equation
The Heat Equation
• A differential equation whose solution provides the temperature distribution in a
stationary medium.
• Based on applying conservation of energy to a differential control volume
through which energy transfer is exclusively by conduction.
• Cartesian Coordinates:
Net transfer of thermal energy into the
control volume (inflow-outflow)
p
T T T T
k k k q c
x x y y z z t

          
                 
(2.19)
Thermal energy
generation
Change in thermal
energy storage

7. 7
qy
qx
qy+dy
qz
qx+dx
qz+dz
x
y
z
 
t
T
dxdydzc
q
q
q
dxdydzq
q
q
q
dzz
dyy
dxx
z
y
x











































)(0

heat in int generation heat out accumulation

8. 8
 
t
T
dxdydzc
q
q
q
dxdydzq
q
q
q
dzz
dyy
dxx
z
y
x











































)(0

 
x
T
dydzkqx


 dx
x
q
qq x
xdxx



 
y
T
dxdzkqy


 dy
y
q
qq
y
ydyy



 
z
T
dxdykqz


 dz
z
q
qq z
zdzz



heat in int generation accumulation
The terms are defined as follows
heat out

9. 9
t
T
cq
z
T
k
zy
T
k
yx
T
k
x 































0

t
T
k
q
z
T
y
T
x
T












10
2
2
2
2
2
2

t
T
k
q
T




102 
2
2
2
2
2
2
2
zyx 








c
k

 
Upon substitution
General heat conduction equation
The Fourier-Biot equation (THE EQUATION
GENERALLY USED)
is the Laplacian operator; is the thermal diffusivity
For isotropic materials (k is constant in all directions)
or

10. 10
Special Cases of the Fourier-Biot equation
00
2
2
2
2
2
2









k
q
z
T
y
T
x
T 
02
2
2
2
2
2









z
T
y
T
x
T
t
T
z
T
y
T
x
T












1
2
2
2
2
2
2
(1) Steady state (Poisson Equation)
Steady Heat Conduction in Plane Walls
There is no heat transfer where there is no change in temperature.
Energy Balance for the Plane Wall
(2) Steady state and no heat generation or sink (Laplace Equation)
(3) Transient (time varying) with no heat generation or sink (Diffusion Equation)
• In the above k is assumed to be constant.
• If k varies with temperature, then k cannot be moved in the equations.

11. 11
Conditions for special cases:
0


t
T
0





zy
00 q
0



0


z
0






a) steady state
b) one- dimensional heat flow
c) no internal generation
d) axi-symmetric problems
e) long cylinder
f) radial heat transfer (spherical)

12. Heat Equation (Radial Systems)
2
1 1
p
T T T T
kr k k q c
r r r z z tr

 
          
                 
(2.26)
• Spherical Coordinates:
• Cylindrical Coordinates:
2
2 2 2 2
1 1 1
sin
sin sin
p
T T T T
kr k k q c
r r tr r r
 
    
          
                 
(2.29)

13. Heat Equation (Special Case)
• One-Dimensional Conduction in a Planar Medium with Constant Properties
and No Generation
2
2
1T T
tx 
 


2
thermal diffusivit of the medium m /sy
p
k
c


  
 
p
T T
k c
x x t

   
 
   
becomes

14. Boundary Conditions
Boundary and Initial Conditions
• For transient conduction, heat equation is first order in time, requiring
specification of an initial temperature distribution:    0
0t=
T x,t =T x,
• Since heat equation is second order in space, two boundary conditions
must be specified. Some common cases:
Constant Surface Temperature:
 0 sT ,t =T
Constant Heat Flux:
0x= s
T
-k | = q
x



Applied Flux Insulated Surface
0 0x=
T
| =
x


Convection:
 0 0x=
T
-k | = h T -T ,t
x


  

20. Example 2.1

21. Example 2.2

22. Example 2.3

23. Example 2.4

24. Example 2.5

25. Example 2.6